Demand and supply fix the prices of traded assets in every market. The prices are taken as given and plain-vanilla derivatives traders view them through the Black-Scholes model to derive implied volatilities. The role of the quants is to create models that are consistent with market data. These models are based on parameters that are local and the task is to make sure the implied parameters by the model match the implied parameters from the market. This is a guarantee of a low model error. The Dupire model as it is explained in [1] is an illustration of a model that uses local volatility to price consistently with European prices or implied volatilities.
The same task can be undertaken for correlation. Unlike the volatility market there is no liquid market with information on implied correlation. However, one-day there will be and firms have to make sure that whatever local correlation, which is put into the model we are able to compute the implied correlation given by the model. We shall work with a model that is consistent with term structure local volatility to explain a term structure of implied volatility. We shall also compute the implied correlation that is the result of a local correlation within this framework. Finally, we raise the question of the right measure of risk when we model dependence.
Framework
We consider two assets that have the following dynamics:
This model needs a local volatility and a local correlation to be specified. However, these parameters generate implied parameters that we are going to calculate.
Implied Parameters
The implied volatility in the previous model is given by the following formula:
Now more importantly the implied correlation is equal to the following quantity:
To compute the previous expression, we compute the covariance between the two assets and divide them by the variances of the assets. It is important to notice that a constant local correlation generates a term structure of implied correlation.
Impact On trading
Trading correlation sensitive products in a global portfolio has to be done carefully. Consider two spread positions on the FTSE and SPX maturing in two years and five years. Imagine that you are long the two year and short the five year. Then to correctly aggregate the risks in your portfolio you cannot use the same local correlation to price the two positions. Instead you have to decide what is the implied correlation either by inferring it in one of the positions or by estimating it through statistical analysis. Then, you make sure that local correlation is such that the implied correlations in the two positions match.
Correlation Or Dependence
In a normal return world, knowing the correlation of the assets is sufficient to know the whole structure of the joint law. This, however, fails in reality as we are dealing with skewed distribution with high peaks. The easy way to create dependence between the different assets by introducing copula techniques. This was the norm in the credit derivatives market. Take two assets, gaussialise their returns and then you can find some correlation between these assets and this will be sufficient to determine the complete distribution of the two asset returns. In a historical point of view, we can estimate the correlation of the two assets by computing the estimated correlation on the modified gaussialise assets.
Modelling For P&L
When we lie in the bath we don't need to understand the molecular structure of water, we just need to understand fluids. In Finance, and especially in derivatives, we do not need to know exactly what the process of finding the spot price is if ultimately we need a model to hedge a call option.
In fact, we have to model the underlying to a certain level that is consistent with the use of it. We should understand what modifies our P&L and so decide how to choose the adequate risk measures.
Conclusion
We derived all the Black-Scholes equations under the real probability. We insisted on the difference between local and implied parameters. Finally, as far as the P&L is concerned, the risk measure that computes the impact of dependence is the covariance rather than the correlation. The same approach can easily be extended to price the fair value of derivatives under the discrete time hypothesis but also price derivatives when volatility hedging is required in addition to delta hedging.
References
[1] Dupire, B. (1994), "Pricing With A Smile", Risk.
This week's Learning Curve was written by Frederic Hatt, exotic equity derivatives trader at Dresdner Kleinwort Wasserstein, and Adil Reghai, an equity derivatives quantitative analyst at DrKW.
